De Morgan's Laws (Set Theory)/Set Complement/Complement of Union/Proof 2

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Theorem

$\overline {T_1 \cup T_2} = \overline T_1 \cap \overline T_2$


Proof

\(\ds \) \(\) \(\ds x \in \overline {T_1 \cup T_2}\)
\(\ds \) \(\leadstoandfrom\) \(\ds x \notin \paren {T_1 \cup T_2}\) Definition of Set Complement
\(\ds \) \(\leadstoandfrom\) \(\ds \neg \paren {x \in T_1 \lor x \in T_2}\) Definition of Set Union
\(\ds \) \(\leadstoandfrom\) \(\ds \neg \paren {x \in T_1} \land \neg \paren {x \in T_2}\) De Morgan's Laws (Logic): Conjunction of Negations
\(\ds \) \(\leadstoandfrom\) \(\ds x \in \overline {T_1} \land x \in \overline {T_2}\) Definition of Set Complement
\(\ds \) \(\leadstoandfrom\) \(\ds x \in \overline {T_1} \cap \overline {T_2}\)

By definition of set equality:

$\overline {T_1 \cup T_2} = \overline {T_1} \cap \overline {T_2}$

$\blacksquare$


Sources


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